Prof. Dr. Timothy Williamson

Barcan Formulas in Second-Order Modal Logic

On the natural extension of Kripke-style possible worlds semantics to second-order modal logic, the Barcan schema and its converse are valid for the second-order quantifiers even though they are invalid for the first-order quantifiers.

I will argue that this creates a metaphysical puzzle, which is no mere artifact of a particular style of formal semantics. Rather, natural comprehension principles for the second-order quantifiers cohere best with a metaphysical conception on which the Barcan schema and its converse hold for both first-order and second-order quantifiers.